## Abstract

We propose a scheme of photon blockade in a system comprising of coupled cavities embedded in Kerr nonlinear material, where two cavities are driven and dissipated. We analytically derive the exact optimal conditions for strong photon antibunching, which are in good agreement with those obtained by numerical simulations. We find that conventional and unconventional photon blockades have controllable flexibilities by tuning the strength ratio and relative phase between two complex driving fields. Such unconventional photon-blockade effects are ascribed to the quantum interference effect to avoid two-photon excitation of the coupled cavities. We also discuss the statistical properties of the photons under given optimal conditions. Our results provide a promising platform for the coherent manipulation of photon blockade, which has potential applications for quantum information processing and quantum optical devices.

© 2015 Optical Society of America

## 1. Introduction

The generation of nonclassical states of photons [1, 2] represents a hot research topic in integrated quantum technologies and quantum optics and is vital for furture applications. In particular, the single-photon source plays an important role in quantum information technologies, and makes their production becoming an ultimate goal of defining a photonic-based architecture in quantum communication [3], quantum metrology [4], and quantum information technologies [5, 6]. The physics behind this mechanism is photon blockade effect, which has been demonstrated in an optical cavity coupled to a quantum emitter [7–10] and coupled single quantum dot-cavity system [11]. Subsequently, a sequence of experimental groups observed the strong antibunching behaviors in different systems, such as a quantum dot in a photonic crystal [12], circuit cavity quantum electrodynamics systems [13, 14].

It is well known that a coherent light beam enters a nonlinear system whose effective nonlinear response is able to produce a shift of the two-photon resonance that is larger than its linewidth [15], which is named as conventional photon blockade. Typical examples include quantum optomechanical systems [16–26] and cavity quantum electrodynamics [27–36]. A sequence of schemes have been proposed in nanostructured cavities and semiconductor microcavities with second-order nonlinear [37, 38]. Such photon blockade has been applied to transistors [39], interferometers [40], and optical rectifiers [41, 42], to mention a few.

The unconventional photon blockade effect was first predicted [43] in photon molecule system [44, 45], which shows that the presence of two photons in two coupled cavities with weak nonlinearities is prevented by destructive quantum interference between different paths in the nonlinear coupled cavity system [46–52]. Similar effects recently were considered in various models, such as coupled single-mode cavities with second- or third-order nonlinearity [53–58], double asymmetric cavities [59], Gaussian squeezed states [60], optical cavity with a quantum dot [61, 62], coupled optomechanical systems [63, 64], and bimodal coupled polaritonic cavities [65, 66], and so on.

Here, we show that an unconventional photon blockade can be engineered in two coupled cavities filled with nonlinear Kerr mediums, where two cavities are driven and dissipated. The approximate optimal conditions for the detuning Δ and Kerr nonlinearity *U* have been derived when *J* ≫ *κ* and *φ* = 0 in [47]. Here we consider a more general scenario, in which we remove these two constraints, and derive the exact optimal conditions in order to generate strong antibunching photons in the system for all the regime of parameters. There are a total of eight sets of solutions for the optimal detuning since the discriminant changes its sign as the strength ratio and the relative phase vary between two complex driving fields, which can be controlled precisely [67]. Hence exact optimal conditions give a fully description of photon blockade, and holds for all range of parameters. On the contrary, if there are no exact optimal conditions, it is very difficult to experimentally find the all eight optimal regimes of photon blockade. Our results provide a promising platform for the coherent manipulation of photon blockade, which has potential applications for quantum information processing and quantum optical devices.

## 2. Physical model and optimal antibunching conditions

The system under consideration is composed of two tunnel-coupled cavities filled with a Kerrtype nonlinear material. We assume that each cavity to be single mode, described by Bose operator â and $\widehat{b}$, respectively. The nonlinear second-quantized Hamiltonian reads [43, 46, 68]

*J*denotes the tunnel coupling rate,

*ω*and

_{j}*U*eigenfrequency and Kerr nonlinearity strength for cavity

_{j}*j*(

*j*=

*a*,

*b*), respectively. The setup is illustrated in Fig. 1(a), where the driving frequency is denoted by

*ω*, and the complex driving strength denoted by ${F}_{j}{e}^{i{\phi}_{j}}$. The total Hamiltonian including the driving terms is then,

_{L}

_{L}*a*=

*ωa*−

*ω*and Δ

_{L}*b*=

*ωb*−

*ω*denote the detunings of cavity

_{L}*a*and

*b*from two driving fields, respectively. Here we note that the relative phase

*φ*=

*φ*−

_{a}*φ*plays an importance role, which can be found by $\widehat{a}\to \widehat{a}{e}^{i{\phi}_{b}}$ $\widehat{b}\to \widehat{b}{e}^{i{\phi}_{b}}$. This also can be seen from the derived optimal conditions (21)–(27). Considering the dissipations to the system, the density matrix

_{b}*ρ*of the system is governed by the master equation,

*Ĥ*is given by Eq. (3),

*κ*and

_{a}*κ*denote loss rates for the two cavities, respectively. The superoperator is defined by $\mathcal{D}(\widehat{o})\rho =\widehat{o}\rho {\widehat{o}}^{\u2020}-\frac{1}{2}{\widehat{o}}^{\u2020}\widehat{o}\rho -\frac{1}{2}\rho {\widehat{o}}^{\u2020}\widehat{o}$. The equal-time second-order correlation function for mode

_{b}*a*can be given by [69, 70],

We present the variation of
${g}_{a}^{(2)}(0)$ defined by Eq. (5) as a function of *U* and Δ by numerically solving master equation (4) in Figs. 1 and 2. First we discuss the features of Fig. 1(b). We find there are three interesting points at *U* = 1.593 × 10^{−3}*κ*, 2.03 × 10^{−3}*κ*, and 1.18 10^{−3}*κ* marked respectively by *A*, *B*,*C*. Points *A*, *B*, and *C* correspond to different relative phases between two complex driving fields, which are denoted by *φ* = 1.6 rad, 1.8 rad, and 1.4 rad, respectively.

Interestingly, we find that there are two minimum values (see points *P*_{1}-*P*) for
${g}_{a}^{(2)}(0)$ in Fig. 2(a) for given strength ratio *r* = *F _{b}*/

*F*, decay rate

_{a}*κ*=

_{a}*κ*≡

_{b}*κ*, and the relative phase

*φ*. However, ${g}_{a}^{(2)}(0)$ appears four minimum values (see points

*P*

_{3}-

*P*

_{6}) in Figs. 2(b)–2(d) as

*r*and

*φ*vary. Therein points

*P*

_{1}-

*P*

_{4}correspond to unconventional single-photon blockade, where Kerr nonlinearity

*U*is smaller than the mode broadening

*κ*, while points

*P*

_{5}and

*P*

_{6}indicate conventional single-photon blockade, where Kerr nonlinearity

*U*is larger than the mode broadening

*κ*. One may wonder why there are four points in Figs. 2(b)–2(d) appearing for single-photon blockade under the same parameters? In the following, we will answer this question.

Assume that the system is initially prepared in |00〉, and only these levels are occupied due to two weak complex driving fields. Therefore the total sate of the system can be written as [46, 71],

*δ*= Δ

_{j}*−*

_{j}*iκ*/2. Under the weak driving condition, we have |

_{j}*A*

_{00}| ≫ |

*A*

_{10}|, |

*A*

_{01}| ≫ |A

_{20}|, |A

_{11}|, |A

_{02}|. In steady state, $\frac{\partial}{\partial t}{A}_{jk}=0$, the relations between

*A*

_{10}and

*A*

_{01}read,

From Eq. (13)*A*_{10} and *A*_{01} can be written as,

The equal-time second-order correlation function ${g}_{a}^{(2)}(0)$ can be given by substituting Eq. (6) into Eq. (5) in weak driving limit.

The conditions for
${g}_{a}^{(2)}(0)\ll 1$ are derived from Eq. (14) by setting *A*_{20} = 0. Hereafter, we will set *U _{b}* ≡

*U*,

*U*= 0 because

_{a}*U*does not appear in the coupled Eq. (14) after taking

_{a}*A*

_{20}= 0.

For simplicity, in the following we mainly discuss the case of Δ* _{a}* = Δ

*≡ Δ,*

_{b}*κ*=

_{a}*κ*≡

_{b}*κ*,

*F*/

_{b}*F*=

_{a}*r*,

*φ*−

_{a}*φ*=

_{b}*φ*. Complementally, in Appendix A we discuss the general situation of Δ

*≠ Δ*

_{a}*and*

_{b}*κ*≠

_{a}*κ*, and give the optimal conditions for strong photon antibunching.

_{b}Noticing Eq. (15), the conditions for
${g}_{a}^{(2)}(0)\to 0$ are derived from Eq. (14) by setting *A*_{20} = 0, so we arrive at,

The condition for *A*_{02}, *A*_{11}, and *A*_{10} to have nontrivial solutions is that the determinant of the coefficient matrix of Eq. (17) is equal to zero, then we obtain

After a simple calculation, we obtain the equation for the detuning Δ:

where these coefficients take*b*=

*rJ*cos(

*φ*)[

*rJ*sin(

*φ*) − 2

*κ*]/2

*κ*, $c=\frac{{J}^{2}}{12}(3+8{r}^{2})+\frac{{\kappa}^{2}}{12}+\frac{rJ}{12\kappa}[rJ\kappa \mathrm{cos}(2\phi )-4({J}^{2}+{r}^{2}{J}^{2}+{\kappa}^{2})\mathrm{sin}(\phi )]$,

*d*=

*rJ*cos(

*φ*)[

*rJ*(4

*J*

^{2}+ 5

*κ*

^{2}) sin(

*φ*) − 4(1 +

*r*

^{2})

*J*

^{2}

*κ*− 2

*κ*

^{3}]/8

*κ*, $e=\frac{1}{2}{r}^{4}{J}^{4}+\frac{1}{8}(8{r}^{2}-1){J}^{2}{\kappa}^{2}+\frac{1}{16}{\kappa}^{4}+\frac{1}{8}rJ[rJ(4{J}^{2}-3{\kappa}^{2})\mathrm{cos}(2\phi )-4\kappa (3{J}^{2}{r}^{2}-{J}^{2}+{\kappa}^{2})\mathrm{sin}(\phi )]$. The nature of roots for Eq. (19) is mainly determined by its discriminant. When the discriminant with

*m*=

*e*−4

*bd*+3

*c*

^{2}and

*n*= (4

*h*

^{3}−

*hm*−

*g*

^{2}), Eq. (19) under this condition has two real roots,

*j*= 1,2 in Eq. (21). Other parameters take

*v*

_{1}= 2

*rJκ*+ (2

*J*

^{2}−

*κ*

^{2}+ 4Δ

^{2})sin(

*φ*),

*v*

_{2}=

*rJκ*−

*κ*

^{2}sin(

*φ*) + 4Δ

^{2}sin(

*φ*),

*v*

_{3}= −16

*rJκ*Δcos(

*φ*) −

*κ*(

*κ*

^{2}− 12Δ

^{2})cos(2

*φ*),

*h*=

*b*

^{2}–

*c*,.

*g*=

*d*– 3

*bc*+ 2

*b*

^{3}, $\lambda =[\sqrt[3]{-n+\sqrt{-\delta /27}}+\sqrt[3]{-n-\sqrt{-\delta /27}}]/2+h$. Set

*η*= 12

*h*

^{2}–

*m*.

When

if and only if are simultaneously satisfied, Eq. (19) has four real roots,With Eqs. (25) and (26), the optimal Kerr nonlinearity is expressed as,

*k*= 3,4,5,6 in Eqs. (25) and (26). Other parameters take

*v*

_{1}= 2

*rJκ*+ (2

*J*

^{2}−

*κ*

^{2}+ 4Δ

^{2})sin(

*φ*),

*v*

_{2}=

*rJκ*−

*κ*

^{2}sin(

*φ*) + 4Δ

^{2}sin(

*φ*),

*v*

_{3}= −16

*rJκ*Δcos(

*φ*) −

*κ*(

*κ*

^{2}− 12Δ

^{2})cos(2

*φ*), $\beta =arc\mathrm{cos}[-n/\sqrt{|m{|}^{3}/27}]$, ${y}_{1}=\sqrt{\left|m\right|/3}\mathrm{cos}(\beta /3)+h$, ${y}_{2,3}=\sqrt{\left|m\right|/3}\mathrm{cos}(\beta /3\pm 2\pi /3)+h$. Here

*s*= −sgn(

*g*) when all

*y*are positive, otherwise

_{j}*s*= sgn(

*g*).

Here we address that when the optimal conditions (21) and (22) [or (25), (26) and (27)] are simultaneously satisfied, strong antibunching can be obtained, otherwise the system is not in strong antibunching regimes.

Now we begin to discuss the features of Figs. 1(b) and 2 applying the optimal conditions Eqs. (21), (25) and (26). In Fig. 1(b), the discriminant *δ* is less than zero, Δ = 0.1539*κ* or −0.1664*κ* by Eq. (21) when *φ* = 1.6 rad, which explains the point *A* in Fig. 1. Δ = −0.2134*κ* or 0.1161*κ* at *φ* = 1.8κ rad explains point *B* when *δ* < 0. The explanation for the point *C* is the same as point *B*.

In Fig. 2, we plot
${g}_{a}^{(2)}(0)$ as a function of *U* and Δ by numerically solving the master Eq. (4). We find that these optimal points *P*_{1}-*P*_{6} are in good agreement with those obtained by the effective Hamiltonian method. For unconventional single-photon blockade regimes, see Eqs. (21) and (22) [i.e., for (a), point
${P}_{1}:({\mathrm{\Delta}}_{opt}^{(2)},{U}_{opt})=(-0.276\kappa ,-0.00436\kappa )$, point
${P}_{2}:({\mathrm{\Delta}}_{opt}^{(1)},{U}_{opt})=(0.1945\kappa ,0.004453\kappa )$ when *δ* < 0], Eqs. (25)–(27) [i.e., in (b), point
${P}_{3}:({\mathrm{\Delta}}_{opt}^{(6)},{U}_{opt})=(-0.1614\kappa ,0.02705\kappa )$, point
${P}_{4}:({\mathrm{\Delta}}_{opt}^{(4)},{U}_{opt})=(0.4439\kappa ,0.01518\kappa )$ when *δ* > 0]. For the conventional single-photon blockade, Δ* _{opt}* and Δ

*take $({\mathrm{\Delta}}_{opt}^{(3)},{U}_{opt})=(6.9165\kappa ,-4.222\kappa )$ (denoted by point*

_{opt}*P*

_{5}), and $({\mathrm{\Delta}}_{opt}^{(5)},{U}_{opt})(-6.81\kappa ,4.78\kappa )$ (denoted by point

*P*

_{6}), which are shown in the Figs. 2(c) and 2(d), respectively. Although the optimal values can be found by numerically solving the master equation, the analytical optimal values given by the effective Hamiltonian have the following advantages. Firstly, the analytical optimal

*U*and Δ (21)–(27) are convenient to analyze the regime of antibunching, this would help in experiments to find the required parameters. Secondly, by the analytical expressions for

*U*and Δ

_{opt}*, we can find the roles the system parameters playing. Hence exact optimal conditions (21)–(27) give a fully detailed description of photon blockade, and hold for all range of parameters. On the contrary, without these exact optimal conditions, it is very difficult to experimentally find all of the optimal regimes of photon blockade.*

_{opt}Physically, in order to understand the conventional and unconventional single-photon blockade, we sketch the corresponding bare energy levels in Fig. 1(c). This requires that two-photon probability amplitude *A*_{20} must be suppressed via destructive quantum interference paths for *A*_{20} (i)
$|0,0\u3009\stackrel{{F}_{a}{e}^{i{\phi}_{a}}}{\to}|1,0\u3009\stackrel{\sqrt{2}{F}_{a}{e}^{i{\phi}_{a}}}{\to}|2,0\u3009$, (ii)
$|0,0\u3009\stackrel{{F}_{a}{e}^{i{\phi}_{a}}}{\to}|1,0\u3009\stackrel{{F}_{b}{e}^{i{\phi}_{b}}}{\to}|1,1\u3009\stackrel{\sqrt{2}J}{\to}|2,0\u3009$, (iii),
$|0,0\u3009\stackrel{{F}_{b}{e}^{i{\phi}_{b}}}{\to}|0,1\u3009\stackrel{{F}_{a}{e}^{i{\phi}_{a}}}{\to}|1,1\u3009\stackrel{\sqrt{2}J}{\to}|2,0\u3009$, (iv),
$|0,0\u3009\stackrel{{F}_{b}{e}^{i{\phi}_{a}}}{\to}|0,1\u3009\stackrel{\sqrt{2}{F}_{b}{e}^{i{\phi}_{b}}}{\to}|0,2\u3009\stackrel{\sqrt{2}J}{\to}|1,1\u3009\stackrel{\sqrt{2}J}{\to}|2,0\u3009$. Above four paths result in the destructive quantum interference to obtain the strong antibunching in the cavity mode *a*. So the nonlinear interaction strength *U* needed for destructive quantum interference becomes smaller and exactly controlled by tuning the strength ratio and relative phase between two complex driving fields.

## 3. Unconventional single-photon blockade

#### 3.1. Influence of the relative phase φ on photon blockade

In Fig. 3, we show
${g}_{a}^{(2)}(0)$ as a function of *U* and *φ* by numerically solving master Eq. (4), where Δ is given by Eq. (21), *φ* is plotted in units of *π*. In this case, the discriminant *δ* = *m*^{3} − 27*n*^{2} in Eq. (20) is always less than zero when phase *φ* varies. We set
${\mathrm{\Delta}}_{opt}={\mathrm{\Delta}}_{opt}^{(1)}$ for (a), whereas
${\mathrm{\Delta}}_{opt}={\mathrm{\Delta}}_{opt}^{(2)}$ for (b). The red-dashed lines in (a)–(b) are plotted with *U* given by Eq. (22). We notice that *the unconventional single-photon blockade* occurs in Figs. 3(a)–3(b) due to the quantum interference between different pathways shown in Fig. 1(c), where the Kerr nonlinear strength *U* is smaller than the mode broadening *κ*. From Fig. 3, we find the analytical results show an excellent agreement with the numerical simulation.

However, when the discriminant *δ* = *m*^{3} − 27*n*^{2} in Eq. (20) changes its sign as the strength ratio increases. How does it influence the single-photon blockade? In Figs. 4 and 5, we plot
${g}_{a}^{(2)}(0)$ as a function of *U* and *φ*, where the detuning Δ is given by Eqs. (21), (25) and (26). Figs. 4 and 5(a)–5(d) correspond to different Δ* _{opt}*. The red-dashed lines in (a)–(d) are plotted with

*U*given by Eqs. (22) and (27). From Eqs. (21), (25) and (26), we find that there are a total of eight sets of solutions for the optimal detuning when the discriminant

*δ*changes its sign, i.e., $({\mathrm{\Delta}}_{opt}^{(j)},{\mathrm{\Delta}}_{opt}^{(k)})$,

*j*= 1,2 and

*k*= 3,4,5,6, which can be seen from Figs. 4 and 5. We notice that the conventional and unconventional single-photon blockade occurs in (a)–(d), where the Kerr nonlinear strength

*U*is smaller and larger than the mode broadening

*κ*, respectively.

From Fig. 4(a), we find that the optimal regimes of
${g}_{a}^{(2)}(0)$ are not continuous, which derives from non-continuity of Δ* _{opt}*, i.e., it may take
${\mathrm{\Delta}}_{opt}^{(1)}$ or
${\mathrm{\Delta}}_{opt}^{(3)}$ through the change of the sign of

*δ*. Similar observations can be found in Figs. 4 and 5. We find that

*U*is a periodic function of

_{opt}*φ*with period 2

*π*, which can be seen from the expression of

*U*in Eqs. (22) and (27), where

*U*depends on

*φ*via sin(

*φ*), cos(

*φ*), sin(2

*φ*), and cos(2

*φ*).

#### 3.2. Comparison between exact and approximate optimal conditions

We now analyze the characteristics of photon-blockade in the coupled cavity systems by comparing the exact optimal conditions with that from the approximate one reported in [47]. The approximate optimal conditions given by [47] are obtained with these two assumptions

in which the optimal conditions are given byHere we have set *r* = 1/*η* for comparison with [47] due to the different ratio defined in two papers.

Now we discuss the situation with the relative phase *φ* = 0. In Appendix B we analytically prove that three inequalities in Eqs. (23) and (24) can not hold simultaneously for all parameters. Therefore we conclude that the exact optimal detuning Δ has only two real roots, which corresponds to two optimal detuning
${\mathrm{\Delta}}_{opt}^{(1),(2)}$ in Eq. (21). Examining Eqs. (28) and (29), we can summarize the comparison of exact and approximate optimal regimes in Table 1, which shows the validity of regimes for the approximate optimal conditions. In the following, we will discuss the three regimes:

- When
*φ*≠ 0 (regime III in Table 1), the approximate optimal conditions (30) and (31) are not available in this case. In section II, we have removed the constraint of two parameters*φ*and*J*in Eqs. (28) and (29) and derived the exact optimal conditions (21)–(27). Therefore there are a total of eight sets of solutions for the optimal detuning when the discriminant*δ*changes its sign as parameters vary, i.e., $({\mathrm{\Delta}}_{opt}^{(j)},{\mathrm{\Delta}}_{opt}^{(k)})$,*j*= 1,2 and*k*= 3,4,5,6. When detunings Δ take its optimal values (21), (25), and (26), we can numerically find all eight optimal Kerr nonlinear strength*U*, which can be seen from Figs. 4 and 5._{opt}

Based on above key results, we concretely turn to some new aspects of our work in the following.

In order to compare with the approximate results (30) and (31), we set *φ* = 0. In Fig. 6 we plot
${g}_{a}^{(2)}(0)$ as a function of *U* and Δ by solving numerically the master equation (4). We mark the minimum values of
${g}_{a}^{(2)}(0)$ by *r*_{1} − *r*_{4} (denoted by red-stars). See Eqs. (21) and (22) [for Fig. 6(a), point *r*_{1}:
${r}_{1}:({U}_{opt},{\mathrm{\Delta}}_{opt}^{(1)})=(0.496\kappa ,0.252\kappa )$; point
${r}_{2}:({U}_{opt},{\mathrm{\Delta}}_{opt}^{(2)})=(-0.425\kappa ,-0.1001\kappa )$] for *J* = *κ*. When *J* = 20*κ* [Figs. 6(b) and 6(c)], point *r*_{3} takes
$({U}_{opt},{\mathrm{\Delta}}_{opt}^{(2)})=(-0.00861\kappa ,0.612\kappa )$, and point *r*_{4} takes
$({U}_{opt},{\mathrm{\Delta}}_{opt}^{(1)})=(-0.00272\kappa ,2.0597\kappa )$.

For the approximate optimal points *s*_{1} and *s*_{2} (denoted by red-circles) given by (30) and (31) in Fig. 6, point *s*_{1} takes (*U _{opt}*, Δ

*)=(0.0505*

_{opt}*κ*,0.1

*κ*), point

*s*

_{2}takes (

*U*, Δ

_{opt}*)=(0.00253*

_{opt}*κ*,2

*κ*).

We can see from Fig. 6(a) that these two optimal values *r*_{1} and *r*_{2} obtained from (21) and (22) are consistent with the numerical simulations, but the point *s*_{1} obtained from the approximate optimal condition (30) and (31) has serious deviations with the numerical simulations. This difference comes from the assumptions *J* ≫ *κ* failing in this case with *J* = *κ* (regime I in Table 1). Second, examining the regime of large tunnel strength *J* ≫ *κ*, e.g., *J* = 20*κ* (regime II in Table 1), in Figs. 6(b) and 6(c) we find that the result of the approximate optimal point *s*_{2} given by Eqs. (30) and (31) is close to the exact optimal point *r*_{4}. It is important to address that our exact methods give two optimal points *r*_{1} and *r*_{2} (or *r*_{3} and *r*_{4}), while the approximate methods only show one point *s*_{1} (or *s*_{2}). Furthermore, we plot
${g}_{a}^{(2)}(0)$ in Fig. 6(d) (the parameter takes *J* = *κ*) and find that two exact optimal detunings
${\mathrm{\Delta}}_{opt}^{(1),(2)}$ (minimum values of blue and red-dashed lines, which correspond optimal points *r*_{1} and *r*_{2}, respectively) given by (21) are consistent with numerical simulations, while the approximate solutions do not exactly predict optimal detuning (mininum value of pink dashed-dotted line, which corresponds the approximate optimal points *s*_{1}).

Correspondingly, when *φ* ≠ 0 (regime III in Table 1), in this case there are a total of six optimal points corresponding to single-photon blockade in Fig. 2. This also is a new result of our work.

Second, we plot
${g}_{a}^{(2)}(0)$ as a function of *r* and *U* with Δ taking its optimal values in Fig. 7, in this case *J* = 0.8*κ* (regime I in Table 1). We find that red-dashed lines given by (22) in Figs. 7(a) and 7(b) are in good agreement with those obtained by numerical simulations, but red-dashed line given by (31) in Fig. 7(c) is not consistent with those obtained by numerical simulations, which originates from the assumption of strong tunnel strength *J* ≫ *κ*. In addition, we plot
${g}_{a}^{(2)}(0)$ in Fig. 7(d) and find that two exact optimal detunings
${\mathrm{\Delta}}_{opt}^{(1),(2)}$ given by (21) are consistent with numerical simulations at *r* = 0.303, while the approximate solutions do not exactly predict optimal point at *r* = 0.303. To compare with *φ* = 0, we plot
${g}_{a}^{(2)}(0)$ as a function of *U* and *r* in Fig. 8 by numerically solving master equation (4) when *φ* = 2.5rad (regime III in Table 1). We find the analytical results show an excellent agreement with the numerical simulations.

We set same quantity *J* = 10*κ* (regime II in Table 1) in Fig. 9. Clearly, black-dashed line given by the exact optimal Kerr nonlinearity *U* in (22) [Figs. 9(a) and 9(b)] and approximate optimal Kerr nonlinearity in (31) [Fig. 9(c)] are in good agreement with numerical simulations. Similar to Fig. 6(c), the approximate optimal Kerr nonlinearity *U* decided by Eq. (31) only gives one optimal value in Fig. 9(c), but it can not predict another optimal value given by exact optimal value in Fig. 9(a). Further observation can be found in Fig. 9(d).

Finally, we let *J* continuously varies with the strength ratio *r* = 0.2 in Fig. 10 (regimes I and II in Table 1). We find that good agreement between the two approaches at large tunnel strength *J* ≫ *κ*, but in this case the result obtained by the approximate optimal conditions (30) and (31) does not work well at small *J*, where the red-dashed in Fig. 10(c) does not follow the exact optimal conditions. Further observation can be found in Fig. 10(d).

The same quantity *φ* = 2.5rad (regime III in Table 1) is taken in Figs. 11 and 12 comparing with Fig. 10, in which we study the influence of the tunneling rate *J* on the single-photon blockade effect as a function of *U* and *J*, where the detuning Δ is given by Eqs. (21), (25) and (26). The discriminant *δ* = *m*^{3} − 27*n*^{2} in Eq. (20) changes its sign when the tunnel strength *J* varies, this results in the changes of solutions to Eq. (19). Figure (a)–(d) are for different Δ* _{opt}*. The white-dashed lines in (a)–(d) are plotted with

*U*given by Eqs. (22) and (27). We find the analytical results show an excellent agreement with the numerical simulations.

By the analytical expressions (21)–(27) for *U _{opt}* and Δ

*, we can find the roles the system parameters play. Also they are used as a bench mark to check the approximate methods and to define their range of validity. For example, applying the exact optimal analytical conditions (21)–(27), we can check the validity of the approximate optimal conditions (30) and (31).*

_{opt}Hence the exact optimal conditions (21)–(27) give a fully description of photon blockade, and holds for all range of parameters. On the contrary, if there are no exact optimal conditions (21)–(27), it is very difficult to experimentally find the all eight optimal regimes of photon blockade.

## 4. Analytical result of the equal-time second-order correlation function

Under the weak driving condition, we have |*A*_{00}| ≫ |*A*_{10}|, |*A*_{01}| ≫ |*A*_{20}|, |*A*_{11}|, |*A*_{02}|. Thus we assume that the vacuum state is approximately occupied with *A*_{00} ≈ 1.

Therefore, we obtain the analytical expression of ${g}_{a}^{(2)}(0)$ by substituting Eq. (44) in Appendix C into Eq. (16)

*q*

_{1}= 16

*κ*

^{2}Δ

^{2}+ (4

*J*

^{2}+

*κ*

^{2}− 4Δ

^{2})

^{2},

*q*

_{2}= [4

*r*

^{2}

*J*

^{2}

*κ*− 8

*rJκ*(

*U*+ 2Δ)cos

*φ*+

*κ*(8Δ

*U*+ 12Δ

^{2}−

*κ*

^{2})cos(2

*φ*) − 4

*rJ*(

*κ*

^{2}− 4Δ

*U*− 4Δ

^{2})sin

*φ*+ 2(

*κ*

^{2}

*U*+ 3

*κ*

^{2}Δ − 2

*J*

^{2}

*U*− 4Δ

^{2}

*U*− 4Δ

^{3})sin(2

*φ*)]

^{2},

*q*

_{3}= [4

*J*

^{2}

*U*+ 8Δ

^{2}(

*U*+ Δ) − 2

*κ*

^{2}(

*U*+ 3Δ)]cos(2

*φ*) + 4

*rJ*(

*U*+ 2Δ)(

*rJ*− 2

*κ*sin

*φ*) +

*κ*[4Δ(2

*U*+ 3Δ) −

*κ*

^{2}]sin(2

*φ*),

*t*

_{1}= [2

*κ*

^{2}(

*U*+ 3Δ) − 8Δ

^{2}(

*U*+ Δ) + 4

*J*

^{2}(

*U*+ 2Δ)]

^{2}, and

*t*

_{2}= [(2Δcos

*φ*+

*κ*sin

*φ*− 2

*rJ*)

^{2}+ (

*κ*cos

*φ*− 2Δsin

*φ*)

^{2}]

^{2}. We plot ${g}_{a}^{(2)}(0)$ as functions of the detuning Δ in Fig. 13. Noticing that the discriminant

*δ*=

*m*

^{3}− 27

*n*

^{2}in Eq. (20) changes its sign so that the solutions of Eq. (19) vary with

*φ*. Therefore, there are a total of six optimal points corresponding to single-photon blockade, which are in good agreement with these points obtained by numerically solving the master Eq. (4) (see the minimum values in Figs. 13(a)–13(f).

## 5. Analytical expression for time-delayed second-order correlation function

In this section, we will investigate the dynamical evolution of the time-delayed second-order correlation function, which can be defined as [69, 72]

*τ*=

*t*

_{1}−

*t*,

*t*→ ∞. We now derive the analytical expression for the time-delayed second-order correlation function

In Fig. 14,
${g}_{a}^{(2)}(\tau )$ is plotted as a function of the time-delay *τ*. We find
${g}_{a}^{(2)}(\tau )=0$ at *τ* = 0, which corresponds to single-photon blockade because *U* and Δ take its optimal values. As *τ* increases
${g}_{a}^{(2)}(\tau )>0$, i.e.,
${g}_{a}^{(2)}(\tau )$ increase above its initial value at finite delay. We thus have
${g}_{a}^{(2)}(\tau )>{g}_{a}^{(2)}(0)$. This is a violation of the classical inequality [31, 71]. This indicates that photons emitted at different times prefer to stay together comparing with initial delayed point. We find from Eq. (34) that when |Δ − *J*| ≫ *κ*/2, a fast oscillation behavior occurs for
${g}_{a}^{(2)}(\tau )$ as the delay time increases at short times. The magnitude of these osciations decreases as *τ* is increased and
${g}_{a}^{(2)}(\tau )$ approaches unity as *τ* → ∞. This explains Figs. 14(a)–14(f).

Interestingly, in Figs. 15(a), 15(b), and 15(d), we observe that there exists so called “beat frequency” phenomenon for photon. Let’s demonstrate the simple case at *ς*_{1} ≈ *ς*_{2} = *ς*. Therefore from two sine waves of amplitude in Eq. (34), we can obtain,

*J*≫ Δ [see Figs. 15(a) and 15(b)], the frequency of the cosine of the expression above, that is Δ, is often too slow to be perceived as a pitch. Instead, it denotes a periodic variation of the sine, whose frequency is

*J*. Therefore, the periodic of beat frequency for photon is Similar, when

*J*≪ Δ [see Fig. 15(d)], beat frequency phenomenon also can appears, whose periodic can be written as However, if the two frequencies are quite close,

*J*~ Δ, We can not observe beat frequency phenomenon [see line

*L*

_{1}in Fig. 15(c)]. It is very important to point out that the suppression of the beat can also be realized even for weak Kerr nonlinearity

*U*≪

*κ*[see lines

*L*

_{2}and

*L*

_{3}in Fig. 15(c)]. Therefore, this may provide us a way to observe the phenomenon of photon blockade through time-delayed second-order correlation function measurements [13, 73]. In experiments we may clearly can measure time-delay second-order correlation function by measuring the beat frequency. Moreover, using this relation between beat frequency and correlation function, a bunch of other applications, such as observation [74] and control [75] of quantum beating, resonant light scattering [76], and so on, have been realized in recent years. Therefore, this scheme we obtained here is not only feasible but also extendable.

## 6. Conclusions

In summary, we have studied unconventional photon-blockade effects in two linear coupled nonlinear cavities filled with nonlinear Kerr mediums with cavity modes being driven and dissipated. By analytical calculations, we derive a condition for optimized photon antibunching. We find the optimal detuning for strong photon antibunching depends on the strength ratio and the relative phase between two complex driving fields. There are a total of eight sets of solutions for the optimal detuning since the discriminant changes its sign as parameters vary. This provides us with a way to control exactly the single-photon blockade and alter it from conventional to unconventional regimes. We find that the analytical expressions of the equal-time and time-delayed correlation function are consistent with numerical simulations. Under the optimal parameters, the system can be used to generate single-photon sources.

This work shows the usefulness of the nonlinear coupled cavities system as single-photon sources on demand, with potential applications in the coherent manipulation of photon blockade with quantum photonic circuits [77]. We thus believe it is worth exactly controlling the single-photon blockade as a promising alternative to ongoing efforts in developing integrated single-photon sources by making use of quantum emitters, such as single molecules, quantum dots, defects in solids and so on.

## Appendix A The optimal conditions for the situation of Δ_{a} ≠ Δ_{b} and *κ*_{a} ≠ *κ*_{b}

_{a}

_{b}

_{a}

_{b}

In this section, we cancel the constraints of the parameters Δ* _{a}* = Δ

*and*

_{b}*κ*=

_{a}*κ*, and analytically give the optimal conditions for the situation of Δ

_{b}*≠ Δ*

_{a}*and*

_{b}*κ*≠

_{a}*κ*. Therefore the condition for

_{b}*A*

_{02},

*A*

_{11}, and

*A*

_{10}in Eq. (17) to have nontrivial solutions is that the determinant of the coefficient matrix of Eq. (17) is equal to zero, then Eq. (18) becomes

First, we select Δ* _{a}* and

*U*as optimal parameters. After a simple calculation to Eq. (38), we obtain a general quadratic equation for the optimal detuning Δ

*,*

_{a}Based on this optimal detuning Δ* _{a,opt}*, then the optimal Kerr nonlinearity can be expressed as

Second, we select Δ* _{b}* and

*U*as optimal parameters. Similarly, Eq. (38) becomes to,

The solutions of for Δ* _{b}* in Eq. (42) have been given by Eqs. (21), (25), and (26) with the replacement

*b*→

*b*

_{1},

*c*→

*c*

_{1},

*d*→

*d*

_{1}, and

*e*→

*e*

_{1}. With the optimal detuning Δ

*, the optimal Kerr nonlinearity*

_{b,opt}*U*can be given by Eq. (41), where Δ

_{b,opt}*= Δ*

_{b}*. Therefore, for realistic experiment with the system parameters Δ*

_{b,opt}*≠ Δ*

_{a}*and κ*

_{b}*≠ κ*

_{a}*, we can exactly control the single-photon blockade by using the optimal parameters from Eq. (39)–(42).*

_{b}## Appendix B The proof of Eq. (19) only having two real roots when *φ* = 0

When the relative phase equals zero, the coefficients *δ*, *h* and *η* in Eqs. (23) and (24) can be reduced to

*N*

_{1}= −162

*J*

^{6}

*k*

^{2}(

*r*

^{2}−1)

^{2}(1+3

*r*

^{2}) − 207

*J*

^{4}

*k*

^{4}(

*r*

^{2}−1)

^{2}+ 64

*κ*

^{8},

*N*2 = −216

*r*

^{2}(

*r*

^{2}−1)

^{2}(1+

*r*

^{2}). When the discrimination

*δ*> 0 in Eq. (23), if and only if two inequalities (24) are simultaneously satisfied, Eq. (19) has four real roots, otherwise it has four conjugate complex roots. We now solve the three inequalities in Eqs. (23) and (24), we find three inequalities can not hold simultaneously for all parameters. Therefore, there are nonphysical optimal values when discriminant

*δ*> 0 at

*φ*= 0. We conclude that the exact optimal detuning Δ has only real roots ${\mathrm{\Delta}}_{opt}^{(1),(2)}$ in Eq. (21) at

*φ*= 0.

## Appendix C Analytical expression of the probability amplitude

We obtain the analytical expression for the equal-time second-order correlation function by solving the steady state in Eqs. (7)–(12) in weak driving limit

*w*

_{1}= (

*κ*+ 2

*i*Δ)[

*κ*+ 2

*i*(

*U*+ Δ)],

*w*

_{2}= 4

*r*

^{2}

*J*

^{2}[

*κ*+

*i*(

*U*+ 2Δ)],

*w*

_{3}= 4

*iJ*

^{2}

*U*−(

*κ*+ 2iΔ)

^{2}[

*κ*+ 2

*i*(

*U*+ Δ)],

*w*

_{4}= 4

*J*

^{2}+ (

*κ*+ 2

*i*Δ)

^{2},

*w*

_{5}=

*κ*+ 2

*i*(

*U*+ Δ).

## Appendix D Analytical derivation of the time-delayed second-order correlation function

Applying the Born approximation and assuming the environment being in zero temperature to Eq. (33), we can obtain the analytical expression for time-delayed two-order correlation function [38],

with*ρ*is the vacuum state of the environment. The dynamical evolution for Eqs. (46) and (4) are equal, i.e., $\dot{\tilde{\rho}}(\tau )\equiv \dot{\rho}(\tau )$ but they have different initial state. Therefore under effective Hamiltonian approximation, $\tilde{\rho}(\tau )=|\tilde{\mathrm{\Psi}}(\tau )\u3009\u3008\tilde{\mathrm{\Psi}}(\tau )|$ and

_{E}*ρ*(

*τ*) = |Ψ(

*τ*)〉〈Ψ(

*τ*)| are equivalent to $|\dot{\tilde{\mathrm{\Psi}}}(\tau )\u3009=|\dot{\mathrm{\Psi}}(\tau )\u3009$. Therefore the time-delayed correlation function can be calculated by introducing a initial state with $|\overline{\mathrm{\Psi}}\u3009$ given by the steady state solutions of Eqs. (7)–(12). Under this initial state, we obtain the time-delayed second-order correlation function,

To be specific, the initial state after the annihilation of a photon in the left cavity *a* is

Equation (48) can be obtained by solving Eqs. (7)–(9) for these amplitudes with the initial state (49). Under the weak driving condition, we have ${A}_{00}={\overline{A}}_{10}$. Therefore, Eq. (7) decouples with Eqs. (8) and (9). Defining two quantities

*A*

_{11},

*A*

_{20}and

*A*

_{20}, and the initial values $x(0)=\sqrt{2}{\overline{A}}_{20}+{\overline{A}}_{11}$, $y(0)=\sqrt{2}{\overline{A}}_{20}-{\overline{A}}_{11}$, decided by Eq. (44). The other parameters take

*u*

_{1}=

*i*(

*δ*+

_{a}*J*),

*u*

_{2}=

*i*(

*δ*−

_{a}*J*), ${\lambda}_{1}=-i{F}_{a}{e}^{i{\phi}_{b}}({e}^{i\phi}+r){A}_{00}$, ${\lambda}_{2}=-i{F}_{a}{e}^{i{\phi}_{b}}({e}^{i\phi}-r){A}_{00}$ with

*A*

_{00}≡

*Ā*

_{10}. The solutions of the first order ordinary differential equation (51) are

Noticing Eq. (44), Eq. (34) can be obtained by substituting Eqs. (50) and (52) into Eq. (48), where these time-independent coefficients take *c*_{3} = 64[(−2*aJ* + 2Δcos*φ* + *κ*sin*φ*)^{2} + (*κ*cos*φ* − 2Δsin*φ*)^{2}]^{2}/[16*κ*^{2}Δ^{2} + (4*J*^{2} + *κ*^{2} − 4Δ^{2})^{2}]^{2}, *r*_{1} = 2[−(2*A*_{001}(*J* + Δ) − *A*_{002}*κ*)(*r* + cos*φ*) + (2*A*_{002}(*J* + Δ) + *A*_{001}*κ*)sin*φ*]/[4(*J* + Δ)^{2} + *κ*^{2}], *r*_{2} = −2[(2*A*_{002}(*J* + Δ) + *A*_{001}*κ*)(*r* + cos*φ*) + (2*A*_{001}(*J* + Δ) − *A*_{002}*κ*)sin*φ*]/[4(*J* + Δ)^{2} + *κ*^{2}], *r*_{3} = 2[(2*A*_{001}(*J* − Δ) + *A*_{002}*κ*)(cos*φ* − *r*) + (2*A*_{002}(Δ − *J*) + *A*_{001}*κ*)sin*φ*]/[4(*J* − Δ)^{2} + *κ*^{2}], *r*_{4} = 2[(2*A*_{002}(*J* − Δ) − *A*_{001}*κ*)(cos*φ* − *r*) + (2*A*_{001}(*J* − Δ) + *A*_{002}*κ*)sin*φ*]/[4(*J* −Δ)^{2} + *κ*^{2}].
${x}_{01}=\mathrm{Re}\{x(0)/[{F}_{a}^{2}{e}^{2i{\phi}_{b}}]\}$,
${y}_{01}=\mathrm{Im}\{x(0)/[{F}_{a}^{2}{e}^{2i{\phi}_{b}}]\}$,
${x}_{02}=\mathrm{Re}\{y(0)/[{F}_{a}^{2}{e}^{2i{\phi}_{b}}]\}$,
${y}_{02}=\mathrm{Im}\{y(0)/[{F}_{a}^{2}{e}^{2i{\phi}_{b}}]\}$,
${A}_{001}=\mathrm{Re}\{{\overline{A}}_{10}/[{F}_{a}{e}^{i{\phi}_{b}}]\}$,
${A}_{002}=\mathrm{Im}\{{\overline{A}}_{10}/[{F}_{a}{e}^{i{\phi}_{b}}]\}$, *z*_{1} = *y*_{02} – *r*_{4}, *z*_{2} = *y*_{01} – *r*_{2}, *z*_{3} = *x*_{02} – *r*_{3}, *z*_{4} = *r*_{1} – *x*_{01},
${\varsigma}_{1}=\sqrt{{z}_{1}^{2}+{z}_{3}^{2}}$,
${\varsigma}_{2}=\sqrt{{z}_{2}^{2}+{z}_{4}^{2}}$, *ς*_{3} = *r*_{2} + *r*_{4}, *ς*_{4} = *r*_{1} + *r*_{3},
${\beta}_{1}=\mathrm{arctan}\left(\frac{{z}_{1}}{{z}_{3}}\right)$,
${\beta}_{2}=\mathrm{arctan}\left(\frac{{z}_{2}}{{z}_{4}}\right)$,
${\beta}_{3}=\mathrm{arctan}\left(\frac{{z}_{3}}{{z}_{1}}\right)$,
${\beta}_{4}=\mathrm{arctan}\left(\frac{{z}_{4}}{{z}_{2}}\right)$.

## Acknowledgments

We would like to thank Prof. Yong Li and Prof. Xiao-Qiang Shao for valuable discussions. This work is supported by National Natural Science Foundation of China (NSFC) under grant Nos 11175032, 61475033, 11405008, 11405026 and the Plan for Scientific and Technological Development of Jilin Province (No. 20150520083JH).

## References and links

**1. **L. Davidovich, “Sub-Poissonian processes in quantum optics,” Rev. Mod. Phys. **68**, 127–173 (1996). [CrossRef]

**2. **A. J. Shields, “Semiconductor quantum light sources,” Nat. Photonics **1**, 215–223 (2007). [CrossRef]

**3. **V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. **81**, 1301–1350 (2009). [CrossRef]

**4. **V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics **5**, 222–229 (2011). [CrossRef]

**5. **E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature **409**, 46–52 (2001). [CrossRef] [PubMed]

**6. **P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. **79**, 135–174 (2007). [CrossRef]

**7. **K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature **436**, 87–90 (2005). [CrossRef] [PubMed]

**8. **B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science **319**, 1062–1065 (2008). [CrossRef] [PubMed]

**9. **F. Dubin, C. Russo, H. G. Barros, A. Stute, C. Becher, P. O. Schmidt, and R. Blatt, “Quantum to classical transition in a single-ion laser,” Nat. Phys. **6**, 350–353 (2010). [CrossRef]

**10. **M. Bajcsy, A. Majumdar, A. Rundquist, and J. Vučković, “Photon blockade with a four-level quantum emitter coupled to a photonic-crystal nanocavity,” New J. Phys. **15**, 025014 (2013). [CrossRef]

**11. **J. Tang, W. D. Geng, and X. L. Xu, “Quantum interference induced photon blockade in a coupled single quantum dot-cavity system,” Sci. Rep. **5**, 9252 (2015). [CrossRef] [PubMed]

**12. **A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. **4**, 859–863 (2008). [CrossRef]

**13. **C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov Jr., M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. **106**, 243601 (2011). [CrossRef] [PubMed]

**14. **A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Türeci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. **107**, 053602 (2011). [CrossRef] [PubMed]

**15. **A. Imamoğlu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. **79**, 1467 (1997). [CrossRef]

**16. **P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. **107**, 063601 (2011). [CrossRef] [PubMed]

**17. **A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. **107**, 063602 (2011). [CrossRef] [PubMed]

**18. **J. Q. Liao and C. K. Law, “Correlated two-photon transport in a one-dimensional waveguide side-coupled to a nonlinear cavity,” Phys. Rev. A **82**, 053836 (2010). [CrossRef]

**19. **J. Q. Liao and F. Nori, “Photon blockade in quadratically coupled optomechanical systems,” Phys. Rev. A **88**, 023853 (2013). [CrossRef]

**20. **P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A **87**, 013839 (2013). [CrossRef]

**21. **X. N. Xu, M. Gullans, and J. M. Taylor, “Quantum nonlinear optics near optomechanical instabilities,” Phys. Rev. A **91**, 013818 (2015). [CrossRef]

**22. **W. Z. Zhang, J. Cheng, J. Y. Liu, and L. Zhou, “Controlling photon transport in the single-photon weak-coupling regime of cavity optomechanics,” Phys. Rev. A **91**, 063836 (2015). [CrossRef]

**23. **X. Y Lü, W. M. Zhang, S. Ashhab, Y. Wu, and F. Nori, “Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems,” Sci. Rep. **3**, 2943 (2013). [CrossRef] [PubMed]

**24. **L. Qiu, L. Gan, W. Ding, and Z. Y. Li, “Single-photon generation by pulsed laser in optomechanical system via photon blockade effect,” J. Opt. Soc. Am. B **30**, 1683–1687 (2013). [CrossRef]

**25. **X. Y. Lü, Y. Wu, J. R. Johansson, H. Jing, J. Zhang, and F. Nori, “Squeezed optomechanics with phase-matched amplification and dissipation,” Phys. Rev. Lett. **114**, 093602 (2015). [CrossRef] [PubMed]

**26. **H. Wang, X. Gu, Y. X. Liu, A. Miranowicz, and F. Nori, “Tunable photon blockade in a hybrid system consisting of an optomechanical device coupled to a two-level system,” Phys. Rev. A **92**, 033806 (2015). [CrossRef]

**27. **A. Miranowicz, M. Paprzycka, Y. X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A **87**, 023809 (2013). [CrossRef]

**28. **M. Leib and M. J. Hartmann, “Bose-Hubbard dynamics of polaritons in a chain of circuit quantum electrodynamics cavities,” New J. Phys. **12**, 093031 (2010). [CrossRef]

**29. **M. J. Werner and A. Imamoḡlu, “Photon-photon interactions in cavity electromagnetically induced transparency,” Phys. Rev. A **61**, 011801 (1999). [CrossRef]

**30. **L. Tian and H. J. Carmichael, “Quantum trajectory simulations of two-state behavior in an optical cavity containing one atom,” Phys. Rev. A **46**, R6801 (1992). [CrossRef] [PubMed]

**31. **R. J. Brecha, P. R. Rice, and M. Xiao, “N two-level atoms in a driven optical cavity: Quantum dynamics of forward photon scattering for weak incident fields,” Phys. Rev. A **59**, 2392 (1999). [CrossRef]

**32. **J. Kim, O. Bensen, H. Kan, and Y. Yamamoto, “A single-photon turnstile device,” Nature **397**, 500–503 (1999). [CrossRef]

**33. **I. I. Smolyaninov, A. V. Zayats, A. Gungor, and C. C. Davis, “Single-photon tunneling via localized surface plasmons,” Phys. Rev. Lett. **88**, 187402 (2002). [CrossRef] [PubMed]

**34. **S. Rebić, A. S. Parkins, and S. M. Tan, “Photon statistics of a single-atom intracavity system involving electromagnetically induced transparency,” Phys. Rev. A **65**, 063804 (2002). [CrossRef]

**35. **D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A **76**, 031805 (2007). [CrossRef]

**36. **Y. X. Liu, X. W. Xu, A. Miranowicz, and F. Nori, “From blockade to transparency: controllable photon transmission through a circuit-QED system,” Phys. Rev. A **89**, 043818 (2014). [CrossRef]

**37. **A. Majumdar and D. Gerace, “Single-photon blockade in doubly resonant nanocavities with second-order non-linearity,” Phys. Rev. B **87**, 235319 (2013). [CrossRef]

**38. **H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Quantum optical diode with semiconductor microcavities,” Phys. Rev. A **90**, 023849 (2014). [CrossRef]

**39. **D. E. Chang, A. S. Sorensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. **3**, 807–812 (2007). [CrossRef]

**40. **D. Gerace, H. E. Tureci, A. Imamoḡlu, V. Giovannetti, and R. Fazio, “The quantum-optical Josephson interferometer,” Nat. Phys. **5**, 281–284 (2009). [CrossRef]

**41. **F. Fratini, E. Mascarenhas, L. Safari, J-Ph. Poizat, D. Valente, A. Auffèves, D. Gerace, and M. F. Santos, “Fabry-Perot interferometer with quantum mirrors: nonlinear light transport and rectification,” Phys. Rev. Lett. **113**, 243601 (2014). [CrossRef] [PubMed]

**42. **E. Mascarenhas, D. Gerace, D. Valente, S. Montangero, A. Auffovès, and M. F. Santos, “A quantum optical valve in a nonlinear-linear resonators junction,” Europhys. Lett. **106**, 54003 (2014). [CrossRef]

**43. **T. C. H. Liew and V. Savona, “Single photons from coupled quantum modes,” Phys. Rev. Lett. **104**, 183601 (2010). [CrossRef] [PubMed]

**44. **M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett. **81**, 2582 (1998). [CrossRef]

**45. **Y. P. Rakovich and J. F. Donegan, “Photonic atoms and molecules,” Laser Photonics Rev. **4**, 179–191 (2010). [CrossRef]

**46. **M. Bamba, A. Imamoğlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A **83**, 021802 (2011). [CrossRef]

**47. **X. W. Xu and Y. Li, “Tunable photon statistics in weakly nonlinear photonic molecules,” Phys. Rev. A **90**, 043822 (2014). [CrossRef]

**48. **I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. **85**, 299 (2013). [CrossRef]

**49. **X. W. Xu and Y. Li, “Strongly correlated two-photon transport in a one-dimensional waveguide coupled to a weakly nonlinear cavity,” Phys. Rev. A **90**, 033832 (2014). [CrossRef]

**50. **J. T. Shen and S. H. Fan, “Strongly correlated two-photon transport in a one-dimensional waveguide coupled to a two-level system,” Phys. Rev. Lett. **98**, 153003 (2007). [CrossRef] [PubMed]

**51. **X. W. Xu and Y. Li, “Strong photon antibunching of symmetric and antisymmetric modes in weakly nonlinear photonic molecules,” Phys. Rev. A **90**, 033809 (2014). [CrossRef]

**52. **T. C. H. Liew and V. Savona, “Multimode entanglement in coupled cavity arrays,” New J. Phys. **15**, 025015 (2013). [CrossRef]

**53. **H. Flayac and V. Savona, “Input-output theory of the unconventional photon blockade,” Phys. Rev. A **88**, 033836 (2013). [CrossRef]

**54. **D. Gerace and V. Savona, “Unconventional photon blockade in doubly resonant microcavities with second-order nonlinearity,” Phys. Rev. A **89**, 031803 (2014). [CrossRef]

**55. **H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Tunable photon blockade in coupled semiconductor cavities,” Phys. Rev. A **91**, 063808 (2015). [CrossRef]

**56. **Y. H. Zhou, H. Z. Shen, and X. X. Yi, “Unconventional photon blockade with second-order nonlinearity,” Phys. Rev. A **92**, 023838 (2015). [CrossRef]

**57. **S. Ferretti, V. Savona, and D. Gerace, “Optimal antibunching in passive photonic devices based on coupled nonlinear resonators,” New J. Phys. **15**, 025012 (2013). [CrossRef]

**58. **O. Kyriienko and T. C. H. Liew, “Triggered single-photon emitters based on stimulated parametric scattering in weakly nonlinear systems,” Phys. Rev. A **90**, 063805 (2014). [CrossRef]

**59. **M. Bamba and C. Ciuti, “Counter-polarized single-photon generation from the auxiliary cavity of a weakly nonlinear photonic molecule,” Appl. Phys. Lett. **99**, 171111 (2011). [CrossRef]

**60. **M. A. Lemonde, N. Didier, and A. A. Clerk, “Antibunching and unconventional photon blockade with Gaussian squeezed states,” Phys. Rev. A **90**, 063824 (2014). [CrossRef]

**61. **A. Majumdar, M. Bajcsy, A. Rundquist, and J. Vučković, “Loss-enabled sub-Poissonian light generation in a bimodal nanocavity,” Phys. Rev. Lett. **108**, 183601 (2012). [CrossRef] [PubMed]

**62. **W. Zhang, Z. Y. Yu, Y. M. Liu, and Y. W. Peng, “Optimal photon antibunching in a quantum-dot-bimodal-cavity system,” Phys. Rev. A **89**, 043832 (2014). [CrossRef]

**63. **X. W. Xu and Y. J. Li, “Antibunching photons in a cavity coupled to an optomechanical system,” J. Opt. B: At. Mol. Opt. Phys. **46**, 035502 (2013).

**64. **V. Savona, “Unconventional photon blockade in coupled optomechanical systems,” arXiv:1302.5937 (2013).

**65. **O. Kyriienko, I. A. Shelykh, and T. C. H. Liew, “Tunable single-photon emission from dipolaritons,” Phys. Rev. A **90**, 033807 (2014). [CrossRef]

**66. **T. C. H. Liew and V. Savona, “Quantum entanglement in nanocavity arrays,” Phys. Rev. A **85**, 050301 (2012). [CrossRef]

**67. **H. A. Bachor and T. C. Ralph, *A Guide to Experiments in Quantum Optics* (Wiley VCH, 2004).

**68. **S. Ferretti and D. Gerace, “Single-photon nonlinear optics with Kerr-type nanostructured materials,” Phys. Rev. B **85**, 033303 (2012). [CrossRef]

**69. **A. Verger, C. Ciuti, and I. Carusotto, “Polariton quantum blockade in a photonic dot,” Phys. Rev. B **73**, 193306 (2006). [CrossRef]

**70. **R. Loudon, *The Quantum Theory of Light* (Oxford University, 2003).

**71. **H. J. Carmichael, R. J. Brecha, and P. R. Rice, “Quantum interference and collapse of the wavefunction in cavity QED,” Opt. Commun. **82**, 73 (1991). [CrossRef]

**72. **S. Ferretti, L. C. Andreani, H. E. Türeci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A **82**, 013841 (2010). [CrossRef]

**73. **J. Wiersig, C. Gies, F. Jahnke, M. Asmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature **460**, 245–249 (2009). [CrossRef] [PubMed]

**74. **Z. Y. Ou and L. Mandel, “Observation of spatial quantum beating with separated photodetectors,” Phys. Rev. Lett. **61**, 54 (1988). [CrossRef] [PubMed]

**75. **A. D. Cimmarusti, C. A. Schroeder, B. D. Patterson, L. A. Orozco, P. Barberis-Blostein, and H. J. Carmichael, “Control of conditional quantum beats in cavity QED: amplitude decoherence and phase shifts,” New J. Phys. **15**, 013017 (2013). [CrossRef]

**76. **M. Peiris, K. Konthasinghe, Y. Yu, Z. C. Niu, and A. Muller, “Bichromatic resonant light scattering from a quantum dot,” Phys. Rev. B **89**, 155305 (2014). [CrossRef]

**77. **J. L. ÓBrien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics **3**, 687–695 (2009). [CrossRef]